Intel is a $53-billion-a-year company that enjoys a near monopoly on the computer chips that go into PCs. But when it comes to the data underlying big companies like Facebook and Google, it says it wants to “return power to the people.”

Intel Labs, the company’s R&D arm, is launching an initiative around what it calls the “data economy”—how consumers might capture more of the value of their personal information, like digital records of their their location or work history. To make this possible, Intel is funding hackathons to urge developers to explore novel uses of personal data. It has also paid for a rebellious-sounding website called We the Data, featuring raised fists and stories comparing Facebook to Exxon Mobil.

Intel’s effort to stir a debate around “your data” is just one example of how some companies—and society more broadly—are grappling with a basic economic asymmetry of the big data age: they’ve got the data, and we don’t.

Internet firms like Google and Amazon are concentrating valuable data about consumers at an unprecedented scale as people click around the Web. But regulations and social standards haven’t kept up with the technical and economic shift, creating a widening gap between data haves and have-nots.

“As consumers, we have no right to know what companies know about us. As companies, we have few restrictions on what we can do with this data,” says Hilary Mason, chief data scientist at Bit.ly, a social-media company in New York. “Even though people derive value, and companies derive value, it’s totally chaotic who has rights to what, and it’s making people uncomfortable.”

In February, for instance, legislators in California introduced the first U.S. law to give individuals a complete view into their online personas. The “Right to Know” bill would let citizens of the state demand a detailed report showing all the information about them that companies like LinkedIn or Google had stored, and whom they had shared it with.

That bill quickly got shelved under pressure from lobbyists for technology companies, who called it “unworkable” and financially damaging to Internet firms and said lawmakers don’t understand “how the Internet works.” Some of the data covered in the bill, like a computer’s IP address, or location, is so basic to communication between machines on the Internet that companies admitted they don’t even know where it ends up.

And that’s the wider dilemma: our personal data is inextricably tied to “big data”—those far larger data sets that now power many of the online services we use. If you don’t tell a navigation app where you are, it can’t tell you where to turn, or tell others there’s traffic ahead. One doesn’t work without the other. What’s more, the economic importance of products fueled with personal data is growing rapidly. According to the Boston Consulting Group, as methods for basing transactions on a person’s digital records have spread from banks to retailers and other sectors, the financial value that companies derived from personal data in Europe was $72 billion in 2011. The consultants concluded that “personal data has become a new form of currency.”

Yet that doesn’t mean it’s a currency easily understood or traded on by individuals. Although a few startups have attempted to help individuals monetize their personal facts (see “A Dollar for Your Data”), the truth is that information about people’s identity and habits has financial value mostly in the aggregate. A single user’s value to Facebook, for instance, is only about $5 a year. Mason, the Bit.ly executive, says trying to put a value on one person’s data is like calculating the value of one unmatched shoe. “And here we are talking about sets of millions or billions of shoes,” she says. “I just don’t think that data plays by the economics of any goods we are familiar with.”

Some believe the market may have already found the right economic balance. “It seems like we have a working model where companies own our data and we’re okay with that because of the free stuff, personalization, and convenience we get in return,” says Gam Dias, CEO of First Retail, an e-commerce consulting company. “There’s not a lot I’m going to do with my extra data anyway. I already know who I am and what I want.”

Intel this year judged the questions swirling around personal data important enough to launch a “Data Economy Initiative,” a multiyear study whose goal is to explore new uses of technology that might let people benefit more directly, and in new ways, from their own data, says Ken Anderson, a cultural anthropologist who is in charge of the project.

Anderson, who once helped Apple develop the sliding application bar that appears on Mac computers (after studying how people organized their desks and stacked items on shelves), says Intel believes technology based on personal data may end up in the control of individuals, in much the same way that mainframe computers gave way to PCs. “It doesn’t matter what you look at in terms of technology. Usually, there is this move toward individualization,” he says.

Intel, which has started surveying consumer opinions, has also been supporting efforts like a competition in New York last fall in which developers wrote apps for the elderly and single mothers. It’s also underwriting the National Day of Civic Hacking, an event focused on new uses of municipal data being released by city governments, such as records of health inspections.

It’s too early to say just what kinds of products might result for Intel, Anderson says,. “When you talk about the data economy, it’s really something that doesn’t yet exist,” he says. “There are people who [are] trying to control a lot of your personal data. But that’s not an economy—that’s just profit for one company.”